I suppose as a convert from naruhina I feel somewhat obligated to certainly not defend them but make sure that they are dissected properly.
I certainly can't speak for part 2 since by then I'd come to my senses and started waving my NaruSaku flag. But as far as part 1 goes it's not so much the sheer mass of interaction between them, even when I had my NaruHina flag I laughed at the people who tried to make it sound like they nearly lived together.
Instead it's more an idea of quality over quantity, Naruto had at that point never really opened up to anyone that we knew and here he was doing it to Hinata. Which in my then gullible mind, along with the minds of alot of other people translated to, Hinata can get through the layers to the real Naruto, and being able to do that is what makes a stable relationship.
Fortunately or unfortunately, I'm now on the fortunately side, Naruto never really dug through Hinata's layers. And then came Sakura who managed to do what Hinata had done only it was a mutual diving through the layers.
I hope this brief insight into the mind of a one-time NaruHina fan helps, even if I wonder from time to time what the heck I was thinking back then.
Now if this is what is on the minds of a lot of Naru/Hina fans then even still that's an even weaker reasoning. The reason why i say this is because, lets be honest; Naruto isn't a deep character. Rather Naruto isn't a character that uses deep thinking or reflective thinking a lot. He's more linear in his mindset. He has an adjective and will see that is completed succeed or fail.
Now the reason why i bring this up is because of this. The way i see this scene is this. Naruto probably forgot about the whole encounter after it happened. He probably didn't forget that he talked to Hinata, but he's not going to delve deeper into any emotions that might have arrisen during the conversation. And to be honest, Ikura was the one that first recognized, accepted and openly showed Naruto affection not Hinata.
So it was like Hinata just had her five minutes in glory, but that was it. Naruto was already thinking about winning his fight.
And I'll be honest with you, there isn't anything in that conversation that specifically screams that it was important that Hinata was with him at that very moment. I could honestly see Sakura, replacing her there. I say this only, because Hinata doesn't leave a very strong impact on the Naruto series.